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Properties

Label 2-220-11.6-c2-0-4
Degree $2$
Conductor $220$
Sign $0.653 - 0.757i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.71 + 3.42i)3-s + (0.690 − 2.12i)5-s + (−2.62 − 3.61i)7-s + (7.70 + 23.7i)9-s + (10.9 + 1.15i)11-s + (9.85 − 3.20i)13-s + (10.5 − 7.65i)15-s + (0.409 + 0.133i)17-s + (−15.0 + 20.6i)19-s − 26.0i·21-s − 21.0·23-s + (−4.04 − 2.93i)25-s + (−28.6 + 88.2i)27-s + (−31.7 − 43.7i)29-s + (−3.90 − 12.0i)31-s + ⋯
L(s)  = 1  + (1.57 + 1.14i)3-s + (0.138 − 0.425i)5-s + (−0.375 − 0.516i)7-s + (0.855 + 2.63i)9-s + (0.994 + 0.105i)11-s + (0.757 − 0.246i)13-s + (0.702 − 0.510i)15-s + (0.0241 + 0.00783i)17-s + (−0.790 + 1.08i)19-s − 1.24i·21-s − 0.915·23-s + (−0.161 − 0.117i)25-s + (−1.06 + 3.26i)27-s + (−1.09 − 1.50i)29-s + (−0.126 − 0.388i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.653 - 0.757i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ 0.653 - 0.757i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.34571 + 1.07437i\)
\(L(\frac12)\) \(\approx\) \(2.34571 + 1.07437i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.690 + 2.12i)T \)
11 \( 1 + (-10.9 - 1.15i)T \)
good3 \( 1 + (-4.71 - 3.42i)T + (2.78 + 8.55i)T^{2} \)
7 \( 1 + (2.62 + 3.61i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-9.85 + 3.20i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (-0.409 - 0.133i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (15.0 - 20.6i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 21.0T + 529T^{2} \)
29 \( 1 + (31.7 + 43.7i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (3.90 + 12.0i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (3.37 - 2.45i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-24.6 + 33.9i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 84.6iT - 1.84e3T^{2} \)
47 \( 1 + (-34.2 - 24.9i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-13.5 - 41.8i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (53.6 - 38.9i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (39.7 + 12.9i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 41.3T + 4.48e3T^{2} \)
71 \( 1 + (-1.24 + 3.84i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-9.80 - 13.4i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-49.2 + 15.9i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-60.4 - 19.6i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 30.4T + 7.92e3T^{2} \)
97 \( 1 + (52.9 + 163. i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.35482935414957873774563604295, −10.82559724845409501878879305217, −10.01790776087061005413490089863, −9.252377502118124398257015151933, −8.467882360151508725264371433983, −7.54587049632882690213902995358, −5.85238784156827764058008380381, −4.08487932749044147863016583458, −3.80866903562551400872761308352, −2.04200917807691340003595277053, 1.57425100992281860418893236369, 2.81821486584447587077514974375, 3.86947417664274592853122111408, 6.25772485139886721749279125758, 6.84564717630924764202244128501, 8.002534042273057667805975089658, 8.988181385177101902636841138568, 9.426909680054938921268925226513, 11.09392587044250296080438151884, 12.26064765215225972403097834557

Graph of the $Z$-function along the critical line